Optical communications have revolutionized the telecommunications industry in recent years. The fiber optic medium provides the ability to efficiently transmit high bit rate signals through a low-loss medium. The development of modern high bandwidth techniques, and wavelength division multiplexing (WDM) to permit the simultaneous transmission of multiple high bandwidth channels on respective wavelengths, has enabled a tremendous increase in communications capacity. The last decade has been seen efforts to increase capacity by taking advantage of the fiber optic medium to the maximum extent possible.
Signals transmitted through an optical medium can be affected by PMD, which is a form of signal distortion that can be caused by subtle physical imperfections in the optical fiber. In principle, an optical fiber with a circular core has rotational symmetry, so that there is no preferred direction for the polarization of the light carrying the optical signal. However, during fabrication, jacketing, cabling, and installation, perturbations in the fiber that will distort this symmetry can occur, thereby causing the fiber to “look different” to various optical polarizations. One of the manifestations of this loss of symmetry is “birefringence,” or a difference in the index of refraction for light that depends on the light's polarization. Light signals with different polarization states will travel at different velocities. In particular, there will be two states of polarization (SOPs), referred to as the “eigenstates” of polarization corresponding to the asymmetric fiber. These eigenstates form a basis set in a vector space that spans the possible SOPs, and light in these eigenstates travels at different velocities.
A birefringent optical fiber transporting a modulated optical signal can temporally disperse the resulting optical frequencies of the signal. For example, an optical pulse, with a given optical polarization, can be formed to represent a “1” in a digital transmission system. If the signal is communicated through a medium with uniform birefringence (i.e., remaining constant along the length of the fiber), the SOPs can be de-composed into corresponding eigenstates, thereby forming two independent pulses, each traveling at its own particular velocity. The two pulses, each a replica of the original pulse, will thus arrive at different times at the end of the birefringent fiber. This can lead to distortions in the received signal at the end terminal of the system. In this simple illustrative case, the temporal displacement of the two replicas, traveling in the “fast” and “slow” SOPs, grows linearly with distance.
In a typical optical communications system, birefringence is not constant but varies randomly over the length of the transmission medium. Thus, the birefringence, and therefore, the eigenstate, changes with position as the light propagates through the length of the fiber. In addition to intrinsic changes in birefringence resulting from imperfections in the fabrication processes, environmental effects such as, for example, temperature, pressure, vibration, bending, etc., can also affect PMD. These effects can likewise vary along the length of the fiber and can cause additional changes to the birefringence. Thus, light that is in the “fast” SOP in one section of fiber might become be in the “slow” SOP at another section of the fiber. Instead of increasing linearly with distance, the temporal separations in the pulse replicas eventually take on the characteristics of a random walk, and grow with the square root of the distance. Despite the local variations in the fast and slow states, it is understood that when the fiber as a whole is considered, another set of states can be defined that characterize the PMD for the entire fiber and split the propagation of the signal into fast and slow components. These “principal states” can be imaged (in a mathematical sense) back to the input face, and used as an alternative basis set. Thus, an arbitrary launch SOP will have components in each of the principal states, and distortion will result from the replication of the pulses after resolution into principal states and their differential arrival times. While the physical process is described in the foregoing in a “global” as opposed to “local” sense, the basic impairment is the same; distortion results from the time delay introduced in the pulse replicas.
The above discussion relates to “narrowband” signals, i.e., having a narrow enough bandwidth that the optical properties of the fiber can be characterized as operating at a single wavelength. This is commonly referred to as “first order PMD.” Birefringence, however, can also vary with wavelength, such that each section of fiber may have slightly different characteristics, both in the magnitude and direction of the birefringence. As a consequence, after a long propagation through an optical medium, light from two neighboring wavelengths initially having the same polarization may experience what looks like a fiber with two different characteristics.
Theoretically, PMD can be represented by a Poincare sphere, or “Stokes' space” representation. In this representation, the equations of motion for SOPs and PMD at a given optical frequency are given by:∂s/∂z=β×s  (1a)∂s/∂ω=τ×s  (1b)∂τ/∂z=∂β/∂ω+β×τ  (1c)In these equations (which are in the “representation” space, not “real” space) “β” represents the birefringence of the fiber at position z, “s” represents the SOP of the light at position z, and “τ” represents the PMD. Generally, Eqn. (1a) states that birefringence causes the representation of the SOP to rotate about the “β” axis as light propagates through the fiber. Eqn. (1b) states that, when viewed at a given position (e.g., the fiber output), the system's PMD causes the SOP to rotate about it as a function of optical frequency. In this regard, light launched at a given optical frequency will evolve to an SOP at the output, and if the optical frequency is then changed (but the launch polarization remains the same), the SOP at the output will also begin to rotate about the PMD vector, τ. Eqn (1c) states that the vector characterizing PMD changes along the length of the fiber. The driving term in Eqn (1c), β′=∂β/∂ω, which we refer to as the “specific PMD,” describes the relationship of birefringence to optical frequency. Even for the simplest cases, there is usually a non-zero driving term (and thus PMD) for birefringent fibers. Based on the above, the vector s will suffer infinitesimal rotations about the axis defined by β, and that the rotation axis will change as β changes with distance (and parametrically with time). However, the total evolution of s can be represented by a single, finite rotation based upon Euler's theorem. If the signal bandwidth is large enough to experience these variations, it is commonly referred to as “higher order” PMD. Higher order PMD also leads to pulse distortion as the optical bandwidth of the signal increases. As the bandwidth increases, the input signal can be decomposed into Fourier components, with each propagated in accordance with the equations discussed above, and the components collected at the output. In the narrowband context, for illustrative purposes, the “concatenation rule” represented by the above equations states that the PMD of a given section of fiber can be “imaged” to the PMD at the output through the same transformation that governs birefringence. For a fiber consisting of two sections having respective PMDs τ1 and τ2, and respective rotations of the SOP via finite rotations R1 and R2, the total PMD can be represented by:τ=τ2+R2τ1  (2)This equation states that the final PMD vector is represented by the vectorial sum of the second (i.e. final) section's PMD vector and the first section's PMD vector, but only after that first PMD vector has been rotated by the same rotation operator (R2) that rotates the SOPs propagating at that wavelength. This is shown by noting the rotations by β implied in Eqns. 1a and 1c.
A generalization of Eqn. 2 shows that a similar rule applies for a fiber having multiple sections. Thus, each section of length Δz can be considered as having it's own uniform primitive PMD vector, β′Δz. The PMD of the entire multi-sectioned fiber can be characterized as a vector sum of the transformed primitive PMDs, one for each section, where each PMD primitive vector is transformed by the concatenated rotation of all the sections between it and the output. Since each of these constituent vectors is only a transformed version of its corresponding primitive PMD vector, each has the same length as its primitive vector, but effectively suffers a random rotation (the Euler's theorem equivalent of the concatenated rotations between the section and the output). This process is illustrated in FIG. 1, where for an arbitrary optical frequency ω0, the fiber (hereinafter, the optical fiber will be referred to as optical fiber) 100 is segregated into five independent sections (i.e., A, B, C, D, E), where each section's PMD is represented by a vector (row 102) directly below that section, and these PMD vectors represent a random distribution in magnitude and direction for the respective sections of the optical fiber. Each section's PMD vector (except the last one's) is imaged to the end and is shown on the right side of the figure (at 106) as a primed version of the original. Thus, the PMD vector for section B is propagated through sections C, D, and E, resulting in its output image, vector B′. The PMD for the entire fiber is then computed as the vector sum of these constituents as depicted at 108 in FIG. 1.
Referring now to FIG. 2, the PMD of the same fiber is shown at a slightly different optical frequency, ω0+Δω. In this example, in row 202 the PMD for each section at ω0 (from FIG. 1) is represented by dotted vectors, while the PMD for each section at ω0+Δω is represented by solid vectors. Each primitive vector corresponding to this neighboring frequency (ω0+Δω) is slightly different than the primitive vector for the original frequency ω0. This, by itself, results in a slightly different sum for the total PMD vector at ω0+Δω. However, in addition to slight changes in the primitive vectors, the new optical frequency also causes different rotations in each section, since the birefringence in each section can also be a function of optical frequency. The images for each section are imaged (trajectories 204) to the output at 206, and are slightly different from those depicted in FIG. 1 as shown by the difference at 206 between the solid and dotted arrows. These change more dramatically as the optical frequency changes. In FIG. 2, the total PMD vector 208 at this new optical frequency is shown as a solid arrow, while the PMD vector at ω0 (from FIG. 1) is depicted as a dotted arrow. Thus, the PMD will change in magnitude and direction as a function of the optical frequency, even though the constituent PMD vectors for the sections may be drawn from the same statistical ensemble representing the fiber's properties. In large part, the study of PMD is a study of the properties of the statistics of the vector sum of these images.
Both the magnitude of the PMD vector, called the “differential group delay” or DGD, and the directions of the unit vectors parallel and anti-parallel to the PMD vector, called the “Principal States of Polarization” (PSPs), change with optical frequency. The principal states are orthogonal and thus are on opposite sides of the sphere. The unit vector is usually associated with the slowest mode. Most frequently, it is the DGD which is plotted in discussions of PMD, but variations in the PSPs with optical frequency also can cause distortion in the optical link. The properties of the PMD are therefore going to follow the statistics of the sum of a set of vectors from the sections of the fiber that are chosen from a distribution and then, for the most part, randomly rotated after propagation through the fiber before being summed.
As discussed above, PMD fluctuates with changes in environmental conditions. Even small environmental changes can add perturbations to the birefringence of sections of the fiber and thereby move many of the imaged primitive vectors. This will consequently change the vector sum. It is to be expected that, at least for subtle environmental changes, the major effect is randomization of the individual rotations in each of the sections. However, since the original distribution was already random, the statistical properties of the perturbed fiber are expected to be essentially the same as those of the original fiber.
Based on the above, a “statistical ensemble” of the PMD can be studied. There are three common statistical ensembles from which observations can be made of this random process: (1) an ensemble of identical, in the statistical sense, fibers in which each primitive section of the fiber is drawn from a sample set, (2) the PMD of a specified fiber transmitting light at a particular wavelength over a very long period of time (this implies the existence of environmental perturbations that will cause each section's primitive vector to change significantly over time and thus also sample the distribution), and (3) the PMD of a specific fiber at a given time over wavelength (where it is assumed that the wavelength spread is sufficient to cause each of the primitive vectors to change significantly over that wavelength spread). The “ergodic hypothesis” is that these three statistical ensembles will have the same properties. This can be represented intuitively by a set of imaged primitive vectors in the three cases. A multi-section fiber (representing the fiber of interest) can be constructed by drawing statistical samples from either a distribution that represents the fiber) or, more generally, the distributions appropriate to the section if a length-dependent fiber is analyzed. Referring again to FIG. 1, a statistical ensemble of identical fibers will draw, for each section, an element from the distribution, and that element will be imaged to the end of the fiber. That is, each section's representative will be imaged to the end and, as each element of the ensemble is examined, that section's contribution will be randomly rotated by the succeeding elements for each ensemble element. Similarly, over a long enough period of time, the image from that section will also be rotated in a random way because environmental perturbations over time will likewise cause the image to be rotated in a random manner. Thus, the average over an ensemble and the average of an ensemble element over time will have the same statistics.
For the case of PMD of a specific fiber at a given time averaged over wavelength, essentially the same distribution of primitive PMD vectors exists. The rotations of the fiber sections vary with frequency because of the linear omega term in β (as discussed above, β=Δn(ω)ω)/c) while the direction and magnitude of the birefringence Δn is set by variations in the slow and fast values of the refractive index, Δn=ns−nf. Thus, the distributions at the wavelengths of interest will be substantially the same distributions as from the earlier samples. Although the rotations corresponding to the various sections may change greatly due to phase accumulation from the frequency changes, the net effect is simply another layer of randomization on an already random variable as long as the rotations lead to further randomization. This further randomization occurs if the change in optical frequency is large enough. Thus, the gross features of the statistical nature of PMD can be captured by the statistics of a set of primitive PMD vectors for the fiber sections that are randomly imaged (i.e. rotated) to the end of the fiber and summed.
If, for example, the probability, Pn of a fiber system having unacceptable PMD impairments, for a given channel, is 10−5 (the “natural” PMD outage probability) then from a statistical ensemble of such fiber systems, 10−5 of them will experience outage at any given instant. In a statistical sense, this might mean that an outage of 5 minutes [i.e., 10−5*(1 year)] can occur per year. Given an ensemble of systems that run for infinite duration, one would expect this to be true when calculating the expectation value for the outage. In real systems, however, it is much more likely that a channel will drift into an outage condition and stay in that condition for some time. Thus, there will be long stretches of time in which the system is operational, perhaps some stretches in which the PMD varies rapidly (due to some perturbation) passing from one operational condition to another through an impaired condition, but there will also be situations in which the system remains impaired for long periods of time. Averaged over very long periods of times, the outage may be 5 minutes per year, but in any given year, it is possible for the system to experience outages of less or much greater duration.
Systems designers may be able to tolerate such impairments, for example, by deploying compensators that correct for the PMD impairments by introducing the opposite PMD. This may lead to additional costs in WDM systems, however, as the following example shows. If, for example, in a WDM system that has 100 wavelengths on a single fiber operating over a link with a natural PMD outage probability of 10−5, PMD that is excessive enough to cause impairment is also likely to vary over wavelength and time, and thus place all channels in potential jeopardy. Accordingly, if one channel needs to be protected with a compensator, then all of them must compensated, or in time, they will all fail. The system operator must therefore purchase 100 compensators, one for each wavelength, even though the expected probability that even one of the compensators might be used at any given time is roughly 0.001 (i.e., 100×10−5). The odds are therefore 1000 to one that none of the compensators are needed at any given time, even though all 100 must be deployed because each of them will eventually be needed at some time.
Co-owned Eiselt et al. (Eiselt) U.S. patent application Ser. No. 09/729,954, filed Dec. 1, 2000, the disclosure of which is hereby incorporated herein, proposes a technique for mitigating the effects of PMD that utilizes a plurality of polarization rotators in an optical fiber to rotate the SOP of an optical signal. The Eiselt method is based on the premise that if the system is about to fail, it is statistically in one of the “1000 to 1” cases (i.e., an unlikely “long-shot” condition has occurred). The Eiselt system changes the optical link slightly to effectively “roll the dice again” in an attempt to hit one of the favorable 999 cases. FIG. 3 conceptually depicts an optical medium 300 segregated into a plurality of birefringent sections separated by the subscripted polarization rotators “R” which are under system control. The polarization rotators are weak birefringent elements that can be controlled to change the SOP of the signal in a prescribed manner. The polarization can be rotated as a result of the weak birefringence so that, in effect, another element of the statistical ensemble corresponding to that fiber can be chosen. In FIG. 3, the primitive vectors in row 302 for each of the sections are the same as those in FIG. 1. The polarization rotators rotate each section's primitive vectors to a new direction 304. As discussed above, each of these new primitive vectors is imaged to the end of the fiber as represented in row 306. These images are summed at 308 to represent the total PMD vector for the fiber with the polarization rotators. In this example, the total PMD vector is smaller than the original PMD (108) shown in FIG. 1 (and represented by the dashed line in FIG. 3), but it could have been larger. The rotators provide an extra level of randomization of the image vectors, and thus will lead to a realization of PMD that is another ensemble element of the original statistical ensemble, since each section's contribution is drawn from the same distribution. When a new configuration is needed, the polarization rotators are set to new states, thereby choosing another fiber realization from the ensemble. Thus, the polarization controllers essentially enable the choice of another ensemble member from the ensemble that represents the fiber.
The net effect is that by setting (possibly random) polarization rotations in the fiber in this way, the current transmission line is replaced with another one in the ensemble. If, as in the example above, the natural outage probability is 10−5 and since there are 100 wavelengths to consider, the odds are overwhelming (e.g. 1000:1) that this new transmission line will not have a serious impairment. In Eiselt, an error signal is generated when, by virtue of monitoring the errors in all 100 channels, one of the channels is determined to be suffering a penalty, presumably due to PMD. This error signal can be generated from SONET bytes that are running parity checks, or from diagnostics in the forward error correction (FEC) circuitry that is used on many modern systems. By monitoring each of the FEC diagnostic channels, the system can determine that one of its channels is nearing (or experiencing) an outage, it then resamples the ensemble space by sending commands to change the settings on the R rotators, hopefully landing on one of the ensemble members for which all the channels are operating.